Unsupervised manifold learning of collective behavior.

Collective behavior is an emergent property of numerous complex systems, from financial markets to cancer cells to predator-prey ecological systems. Characterizing modes of collective behavior is often done through human observation, training generative models, or other supervised learning techniques. Each of these cases requires knowledge of and a method for characterizing the macro-state(s) of the system. This presents a challenge for studying novel systems where there may be little prior knowledge. Here, we present a new unsupervised method of detecting emergent behavior in complex systems, and discerning between distinct collective behaviors. We require only metrics, d(1), d(2), defined on the set of agents, X, which measure agents' nearness in variables of interest. We apply the method of diffusion maps to the systems (X, d(i)) to recover efficient embeddings of their interaction networks. Comparing these geometries, we formulate a measure of similarity between two networks, called the map alignment statistic (MAS). A large MAS is evidence that the two networks are codetermined in some fashion, indicating an emergent relationship between the metrics d(1) and d(2). Additionally, the form of the macro-scale organization is encoded in the covariances among the two sets of diffusion map components. Using these covariances we discern between different modes of collective behavior in a data-driven, unsupervised manner. This method is demonstrated on a synthetic flocking model as well as empirical fish schooling data. We show that our state classification subdivides the known behaviors of the school in a meaningful manner, leading to a finer description of the system's behavior.

Identifying multiscale spatio-temporal patterns in human mobility using manifold learning.

When, where and how people move is a fundamental part of how human societies organize around every-day needs as well as how people adapt to risks, such as economic scarcity or instability, and natural disasters. Our ability to characterize and predict the diversity of human mobility patterns has been greatly expanded by the availability of Call Detail Records (CDR) from mobile phone cellular networks. The size and richness of these datasets is at the same time a blessing and a curse: while there is great opportunity to extract useful information from these datasets, it remains a challenge to do so in a meaningful way. In particular, human mobility is multiscale, meaning a diversity of patterns of mobility occur simultaneously, which vary according to timing, magnitude and spatial extent. To identify and characterize the main spatio-temporal scales and patterns of human mobility we examined CDR data from the Orange mobile network in Senegal using a new form of spectral graph wavelets, an approach from manifold learning. This unsupervised analysis reduces the dimensionality of the data to reveal seasonal changes in human mobility, as well as mobility patterns associated with large-scale but short-term religious events. The novel insight into human mobility patterns afforded by manifold learning methods like spectral graph wavelets have clear applications for urban planning, infrastructure design as well as hazard risk management, especially as climate change alters the biophysical landscape on which people work and live, leading to new patterns of human migration around the world.

Simulation of fractional Brownian surfaces via spectral synthesis on manifolds.

Using the spectral decomposition of the Laplace-Beltrami operator, we simulate fractal surfaces as random series of eigenfunctions. This approach allows us to generate random fields over smooth manifolds of arbitrary dimension, generalizing previous work with fractional Brownian motion with multidimensional parameter. We give examples of surfaces with and without boundary and discuss implementation.